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CHAPTER 8
Bonds and Their Valuation
Key features of bonds
Bond valuation
Measuring yield
Assessing risk
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Key Features of a Bond
1. Par value: Face amount; paid at maturity. Assume $1,000.
2. Coupon interest rate: Stated
interest rate. Multiply by par
to get $ of interest. Generally
fixed.
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3. Maturity: Years until bond must be repaid. Declines.
4. Issue date: Date when bond
was issued.
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How does adding a “call provision”
affect a bond?
Issuer can refund if rates decline.
That helps the issuer but hurts the investor.
Therefore, borrowers are willing to pay more, and lenders require more, on callable bonds.
Most bonds have a deferred call and
a declining call premium.
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What’s a sinking fund?
Provision to pay off a loan over its life rather than all at maturity.
Similar to amortization on a term loan.
Reduces risk to investor, shortens average maturity.
But not good for investors if rates decline after issuance.
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1. Call x% at par per year for sinking fund purposes.
2. Buy bonds on open market.
Company would call if k
dis below the coupon rate and bond sells at a
premium. Use open market purchase if k
dis above coupon rate and bond
Sinking funds are generally handled in
2 ways
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Financial Asset Values
PV = CF
1+ k ... + CF
1+k
1 n
1
2
1
2CF
k
n.
0 1 2 n
k
CF1 CF2 CFn
Value
...
+ +
+
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The discount rate (ki) is the
opportunity cost of capital, i.e., the rate that could be earned on
alternative investments of equal risk.
k
i= k
*+ IP + LP + MRP + DRP.
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What’s the value of a 10-year, 10%
coupon bond if k
d= 10%?
VB $100kd $1,kd
1
000
1 . . . + $10010 1 10
1+ kd
100 100
0 1 2 10
10%
100 + 1,000 V = ?
...
= $90.91 + . . . + $38.55 + $385.54
= $1,000.
+
+ +
+
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10 10 100 1000 N I/YR PV PMT FV
-1,000
The bond consists of a 10-year, 10%
annuity of $100/year plus a $1,000 lump sum at t = 10:
$ 614.46 385.54 $1,000.00 PV annuity
PV maturity value PV annuity
=
=
=
INPUTS OUTPUT
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10 13 100 1000 N I/YR PV PMT FV
-837.21
When k
drises, above the coupon rate, the bond’s value falls below par, so it sells at a discount.
What would happen if expected
inflation rose by 3%, causing k = 13%?
INPUTS OUTPUT
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What would happen if inflation fell, and k
ddeclined to 7%?
10 7 100 1000 N I/YR PV PMT FV
-1,210.71
Price rises above par, and bond sells at a premium, if coupon > k .
INPUTS OUTPUT
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The bond was issued 20 years ago and now has 10 years to maturity.
What would happen to its value over time if the required rate of return remained at 10%, or at 13%, or at
7%?
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M
Bond Value ($)
Years remaining to Maturity
1,372 1,211
1,000
837 775
30 25 20 15 10 5 0
kd = 7%.
kd = 13%.
kd = 10%.
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At maturity, the value of any bond must equal its par value.
The value of a premium bond would decrease to $1,000.
The value of a discount bond would increase to $1,000.
A par bond stays at $1,000 if kd remains constant.
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What’s “yield to maturity”?
YTM is the rate of return earned on a bond held to maturity. Also called
“promised yield.”
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What’s the YTM on a 10-year, 9%
annual coupon, $1,000 par value bond that sells for $887?
90 90 90
0 1 9 10
kd=?
1,000 PV1
. . .PV10 PVM
887
Find k
dthat “works”!
...
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10 -887 90 1000 N I/YR PV PMT FV 10.91
V INT
k
M
B k
d
N
d
N
1 1
...
+ INT 11+ kd
887 90
1
1000
1 10 1 10
kd + 90 kd
1+ kd
,
Find k
d+
+ +
+
+ + + +
INPUTS OUTPUT
...
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If coupon rate < k
d, discount.
If coupon rate = k
d, par bond.
If coupon rate > k
d, premium.
If k
drises, price falls.
Price = par at maturity.
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Find YTM if price were $1,134.20.
10 -1134.2 90 1000 N I/YR PV PMT FV
7.08
Sells at a premium. Because coupon = 9% > k
d= 7.08%,
INPUTS OUTPUT
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Definitions
Current yield = .
Capital gains yield = .
= YTM = + . Annual coupon pmt
Current price
Change in price Beginning price Exp total
return Exp
Curr yld Exp cap
gains yld
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Current yield =
= 0.1015 = 10.15%.
Find current yield and capital gains yield for a 9%, 10-year bond when the bond sells for $887 and YTM = 10.91%.
$90
$887
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YTM = Current yield + Capital gains yield.
Cap gains yield = YTM – Current yield = 10.91% – 10.15%
= 0.76%.
Could also find value in Years 1 and 2,
get difference, and divide by value in
Year 1. Same answer.
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What’s interest rate (or price) risk?
Does a 1-year or 10-year 10% bond have more risk?
k
d1-year Change 10-year Change
5% $1,048 $1,386
10% 1,000 +4.8%
-4.4% 1,000 +38.6%
-25.1%
Interest rate risk: Rising k
dcauses
bond’s price to fall.
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0 500 1,000 1,500
0% 5% 10% 15%
1-year 10-year
k
dValue
.
. .
. . .
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What is reinvestment rate risk?
The risk that CFs will have to be
reinvested in the future at lower rates, reducing income.
Illustration: Suppose you just won
$500,000 playing the lottery. You’ll invest the money and live off the
interest. You buy a 1-year bond with a
YTM of 10%.
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Year 1 income = $50,000. At year-end get back $500,000 to reinvest.
If rates fall to 3%, income will drop from $50,000 to $15,000. Had you
bought 30-year bonds, income would
have remained constant.
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Long-term bonds: High interest rate risk, low reinvestment rate risk.
Short-term bonds: Low interest rate risk, high reinvestment rate risk.
Nothing is riskless!
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Do all bonds of the same maturity
have the same price and reinvestment rate risk?
No, low coupon bonds have less
reinvestment rate risk but more price
risk than high coupon bonds.
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True or False: “All 10-year bonds have the same price and
reinvestment rate risk.”
False! Low coupon bonds have
less reinvestment rate risk but
more price risk than high coupon
bonds.
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Semiannual Bonds
1. Multiply years by 2 to get periods = 2n.
2. Divide nominal rate by 2 to get periodic rate = k
d/2.
3. Divide annual INT by 2 to get PMT = INT/2.
2n kd/2 OK INT/2 OK N I/YR PV PMT FV INPUTS
OUTPUT
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2(10) 13/2 100/2
20 6.5 50 1000 N I/YR PV PMT FV
-834.72
Find the value of 10-year, 10% coupon, semiannual bond if k
d= 13%.
INPUTS OUTPUT
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You could buy, for $1,000, either a 10%, 10-year, annual payment bond or an equally risky 10%, 10-year semiannual
bond. Which would you prefer?
The semiannual bond’s EFF% is:
10.25% > 10% EFF% on annual bond, so buy semiannual bond.
%.
25 . 10 2 1
10 . 1 0
m 1 1 i
% EFF
2 m
Nom
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If $1,000 is the proper price for the semiannual bond, what is the proper
price for the annual payment bond?
Semiannual bond has kNom = 10%, with
EFF% = 10.25%. Should earn same EFF%
on annual payment bond, so:
10 10.25 100 1000 N I/YR PV PMT FV
-984.80 INPUTS
OUTPUT
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At a price of $984.80, the annual and semiannual bonds would be in equilibrium, because investors would earn EFF% = 10.25% on
either bond.
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A 10-year, 10% semiannual coupon,
$1,000 par value bond is selling for
$1,135.90 with an 8% yield to maturity.
It can be called after 4 years at $1,050.
What’s the bond’s nominal yield to call (YTC)?
8 -1135.9 50 1050 N I/YR PV PMT FV 3.568 x 2 = 7.137%
INPUTS OUTPUT
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k
Nom= 7.137% is the rate brokers
would quote. Could also calculate EFF% to call:
EFF% = (1.03568)
2– 1 = 7.26%.
This rate could be compared to
monthly mortgages, etc.
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If you bought bonds, would you be more likely to earn YTM or YTC?
Coupon rate = 10% vs. YTC = k
d=
7.137%. Could raise money by selling new bonds which pay 7.137%.
Could thus replace bonds that pay
$100/year with bonds that pay only
$71.37/year.
Investors should expect a call, hence
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In general, if a bond sells at a
premium, then (1) coupon > k
d, so (2) a call is likely.
So, expect to earn:
YTC on premium bonds.
YTM on par & discount bonds.
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Disney recently issued 100-year bonds with a YTM of 7.5%--this represents
the promised return. The expected return was less than 7.5% when the bonds were issued.
If issuer defaults, investors receive less than the promised return.
Therefore, the expected return on
corporate and municipal bonds is less than the promised return.
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Bond Ratings Provide One Measure of Default Risk
Investment Grade Junk Bonds Moody’s Aaa Aa A Baa Ba B Caa C S&P AAA AA A BBB BB B CCC D
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What factors affect default risk and bond ratings?
Financial performance
Debt ratio
TIE, FCC ratios
Current ratios
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Provisions in the bond contract
Secured vs. unsecured debt
Senior vs. subordinated debt
Guarantee provisions
Sinking fund provisions
Debt maturity
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Other factors
Earnings stability
Regulatory environment
Potential product liability
Accounting policies
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Top Ten Largest U.S. Corporate Bond Financings, as of July 1999 Issuer
Ford Motor Co.
AT&T
RJR Holdings WorldCom
Sprint
Date
July 1999 Mar 1999 May 1989 Aug 1998 Nov 1998
Amount
$8.6 billion
$8.0 billion
$6.1 billion
$6.1 billion
$5.0 billion
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Bankruptcy
Two main chapters of Federal Bankruptcy Act:
Chapter 11, Reorganization
Chapter 7, Liquidation
Typically, company wants Chapter 11,
creditors may prefer Chapter 7.
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If company can’t meet its obligations, it files under Chapter 11. That stops
creditors from foreclosing, taking assets, and shutting down the
business.
Company has 120 days to file a reorganization plan.
Court appoints a “trustee” to supervise reorganization.
Management usually stays in control.
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Company must demonstrate in its reorganization plan that it is
“worth more alive than dead.”
Otherwise, judge will order
liquidation under Chapter 7.
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If the company is liquidated, here’s the payment priority:
1. Secured creditors from sales of secured assets.
2. Trustee’s costs
3. Wages, subject to limits 4. Taxes
5. Unfunded pension liabilities 6. Unsecured creditors
7. Preferred stock 8. Common stock
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In a liquidation, unsecured creditors generally get zero. This makes them more willing to participate in
reorganization even though their claims are greatly scaled back.
